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A005430
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Apéry numbers: n*C(2n,n).
(Formerly M2028)
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16
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0, 2, 12, 60, 280, 1260, 5544, 24024, 102960, 437580, 1847560, 7759752, 32449872, 135207800, 561632400, 2326762800, 9617286240, 39671305740, 163352435400, 671560012200, 2756930576400, 11303415363240, 46290177201840
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| sum(n=1,inf,1/a(n))=Pi*sqrt(3)/9 - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002
Appears as diagonal in A003506. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2006
The aerated sequence 1,0,2,0,12,0,60,0,... has e.g.f. 1+x*Bessel_I(1,2x). [From Paul Barry (pbarry(AT)wit.ie), Mar 29 2010]
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REFERENCES
| F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.25).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. J. van der Poorten, A proof that Euler missed...Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
I. J. Zucker, On the series $ Sum\sp \infty\sb {k=1}(\sp{2k}\sb {\; k})\sp {-1}k\sp{-n}$ and related sums, J. Number Theory 20 (1985), no. 1, 92-102.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
H. J. H. Tuenter, Walking into an absolute sum
Wadim Zudilin, An elementary proof of Apery's theorem
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FORMULA
| G.f.: 2x/sqrt((1-4*x)^3). [From Marco A. Cisneros Guevara, Jul 25 2011]
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MAPLE
| A005430 := n->n*binomial(2*n, n);
with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1), size=n)), j=0..n) od: seq(a[n], n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 03 2007
a:=n->add(binomial(2*n, n), k=1..n): seq(a(n), n=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
a:=n->abs(sum((binomial(-n, n-3)), j=2..n)): seq(a(n), n=2..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
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PROG
| (PARI) a(n)=-(-1)^n*real(polcoeff(serlaplace(x^2*besselh1(1, 2*x)), 2*n)) (from R. Stephan)
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CROSSREFS
| Cf. A002736, A005258, A005259, A005429, A005430. 1/beta(n, n+1) in A061928.
a(n) = A002011(n-1)/2 = 2 * A002457(n-1).
Cf. A001803.
Cf. A003506.
Sequence in context: A009618 A143770 A062478 * A094434 A001574 A074445
Adjacent sequences: A005427 A005428 A005429 * A005431 A005432 A005433
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000
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